Question

The value of ${\int }_{-1}^1\left[x\right[1+sin\pi x]+1]dx$ ([.] denotes the greatest integer)

• A. 2
• B. \N
• C. 1
• D. None of these

Answer 1

The correct option is A 2

${\int }_{-1}^1\left[x\right[1+sin\pi x]+1]dx$
$={\int }_{-1}^0\left[x\right(1+sin\pi x)+1]dx+{\int }_0^1\left[x\right[1+sin\pi x]+1]dx$
$Now-1<x<0\Rightarrow [1+sin\pi x]=0$
$0<x<1\Rightarrow [1+sin\pi x]=1$
$\Rightarrow \left[x\right[1+sin\pi x]+1]=2$
$So={\int }_0^1\left[x\right[1+sin\pi x]+1]dx=2$